36 Prof. Maxivell, On a Paradox in the [Mar. 12, 



The problem has been already solved by Green* in a far more 

 general manner, but at the same time by a far more intricate 

 method. 



We have, as before, for corresponding values of x^ and x^ 



A L„=-i 3- (1). 



x^-p a^-p p-^2 P-^i 

 . Transposing 



so^-p p — ci^ P~'^2 cii—p 

 Multiplying 



{x^ - g J (x^ - a,) ^ ^ (a, - x^) {a^ - x^) , 



{x,-pY{a^-p){p-^) {p-cc^\a^-p){p-a^ '■ \ ■ 



If we write («i - a^i) K - ^2) = 3/i W' 



(a^ - a?,) (^, - a J = 2/, (5), 



we find from equation (3) 



x,-p ^ p-x^ ___ (i^y 



Vi 3/2 



Let /Oj, P2 be the densities and s^, s^ the sections of the rod at 

 the corresponding points X^ and X^, and let the repulsion of the 

 matter of the rod vary inversely as the tz"' power of the distance, 

 then the condition of equilibrium of a particle at P under the 

 action of the elements dx^ and — dx^ is 



p^s^dx^ [x^ -py = - p^s^dx^ {p - x^-^ (7). 



Eliminating dx^ and dx^ by means of equation (2), we find 



p^^^{^^-pr''=pMv-^r' (8), 



and from this by means of equation (6) we obtain 



PAyr = p-Ayr {% 



as the condition of equilibrium between the elements. 



The condition of equilibrium is therefore satisfied for every 

 pair of elements by making 



psif'"' = constant = C (10). 



* George Green. Mathematical Investigations concerning the laws of the 

 equilibrium of fluids analogous to the electric fluid, with other similar researches. 

 Transactions of the Camhridge Philosophical Society, 1833. (Eead Nov. 12, 1832.) 

 Ferrers' Edition of Green's Papers, p. 119. 



